Geodesic
The largest Irreducible Character Degree of a Finite Group
Canadian journal of mathematics, Tome 37 (1985) no. 3, pp. 442-451

Voir la notice de l'article provenant de la source Cambridge University Press

Much information about a finite group is encoded in its character table. Indeed even a small portion of the character table may reveal significant information about the group. By a famous theorem of Jordan, knowing the degree of one faithful irreducible character of a finite group gives an upper bound for the index of its largest normal abelian subgroup.Here we consider b(G), the largest irreducible character degree of the group G. A simple application of Frobenius reciprocity shows that b(G) ≧ |G:A| for any abelian subgroup A of G. In light of this fact and Jordan's theorem, one might seek to bound the index of the largest abelian subgroup of G from above by a function of b(G). If is G is nilpotent, a result of Isaacs and Passman (see [7, Theorem 12.26]) shows that G has an abelian subgroup of index at most b(G) 4. 
Gluck, David. The largest Irreducible Character Degree of a Finite Group. Canadian journal of mathematics, Tome 37 (1985) no. 3, pp. 442-451. doi: 10.4153/CJM-1985-026-8
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[1] 1. Dornhoff, L., Group representation theory, Part A (Marcel Dekker, New York, 1971). Google Scholar

[2] 2. Gorenstein, D., Finite groups (Harper and Row, New York, 1968). Google Scholar

[3] 3. Gorenstein, D., Finite simple groups (Plenum Press, New York, 1982). Google Scholar | DOI

[4] 4. Griess, R. L. Jr. and Lyons, R., The automorphism group of the Tits simple group, Proc. A.M.S. 52 (1975), 75–78. Google Scholar

[5] 5. Hall, M. Jr., Combinatorial theory (Blaisdell, New York, 1967). Google Scholar

[6] 6. Huppert, B., Endliche gruppen I (Springer Verlag, Berlin, 1967). Google Scholar | DOI

[7] 7. Isaacs, I. M., Character theory of finite groups (Academic Press, New York, 1976). Google Scholar

[8] 8. Isaacs, I. M., Character correspondences in solvable groups, Advances in Math. 43 (1982), 284–306. Google Scholar

[9] 9. Passman, D. S., Groups with normal solvable Hall p′-subgroups, Trans. A.M.S. 123 (1966), 99–111. Google Scholar

[10] 10. Praeger, C. and Saxl, J., On the orders of primitive permutation groups, Bull. L.M.S. 37 (1980), 303–307. Google Scholar

[11] 11. Wolf, T. R., Solvable and nilpotent subgroups of GL(n, qm), Can. J. Math. 34 (1982), 1097–1111. Google Scholar

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